图像处理的变分模型若干问题研究

发布时间：2018-09-19 06:46

【摘要】：图像是人类获取和传递信息的重要媒介,在网络信息时代中发挥着重要作用。图像处理则是信息科学与工程中的一门快速发展的新兴交叉学科,在社会生活中具有十分重要的地位和应用价值。变分模型是研究求解泛函极值(极大或极小)的数学模型之一,具有灵活的结构、多样的形式以及高效的运算性能等特性,上世纪90年代初期建立起来的图像处理变分模型已经成为数字图像处理领域中的一类重要数学工具,受到国内外学者的广泛研究和关注。本文围绕图像处理中的去噪、修复以及分割等若干关键问题,针对变分模型在图像处理中存在的自适应性能较差、平滑区域容易出现“阶梯效应”等问题,研究了变分模型的特点和意义,并提出改进思路和方法。论文的主要工作如下：(1)在图像去噪的变分模型研究方面,针对Rudin-Osher-Fatemi(简称ROF)模型中的正则项不可微分问题,提出了基于光滑化ROF模型的图像去噪算法。该算法修改了ROF模型中的正则项,不仅可以摆脱Courant-Friedrichs-Lewy条件,还使其具有全局收敛特性。通过采用原始-对偶法求解光滑化ROF模型,每次迭代更新都对应着一个原始变量和一个对偶变量,相比时间行进法和不动点法在稳定性和效率方面都有更好的表现。仿真实验结果表明,当选择合适的光滑参数后,提出的模型能够有效地减少在图像平稳区域上的“阶梯效应”,同时也能保护图像的边缘。(2)在图像修复的变分模型研究方面,首先,在分析了全变分模型的基础上,采用更加合理的势函数,提出了一种改进的图像修复全变分模型。该模型的优点在于可以保持图像的边缘,模型的解只在等照度线方向扩散,而在梯度方向不扩散。如果待修复图像含有噪声,则可以在修复图像的同时对信息缺损区域的外部去除噪声,不仅避免了“阶梯效应”,还保护了图像的重要特征。其次,针对Mumford-Shah模型求解过程比较繁琐的问题,提出了一种基于凸Mumford-Shah模型的图像修复算法。该算法充分考虑二次范数和全变分范数在图像纹理区域和平滑区域不同的扩散能力,避免了求解曲线长度项的困扰,并采用分裂Bregman算法对模型进行快速求解。仿真实验结果表明,该算法运算效率较高,并且能够保持图像边缘曲线的光滑性。(3)在图像分割的变分模型研究方面,首先,针对Mumford-Shah模型计算复杂度高的问题,提出了一种基于约束Mumford-Shah模型的图像分割方法。该算法通过修改Mumford-Shah模型中的长度项,将非凸的Mumford-Shah模型转化为凸优化问题,并在迭代过程中始终将图像的像素值约束在合理的范围内,然后采用交替方向乘子法对模型进行求解。在获得模型的光滑解之后,采用K-均值聚类法对图像进行分割,实现了光滑和分割的同步进行,运算效率更高,并具有自适应多重分割的性能。其次,考虑凸Mumford-Shah模型中的系数矩阵是随机矩阵时,交替方向乘子法的运算效率下降问题,提出了基于非精确交替方向乘子法的图像分割算法。该算法的优点在于不仅能够处理噪声和模糊图像的分割问题,还可处理随机采样图像的分割问题。该策略的另一个优势是若改变分割目标的个数,则不用重新计算图像的光滑解,应用起来更加方便。仿真实验结果表明,该策略能够处理退化图像的图像分割问题,并且具有较高的分割精度。
[Abstract]:Image is an important medium for human to obtain and transmit information and plays an important role in the network information age. Image processing is a rapidly developing interdisciplinary subject in Information Science and engineering. It has a very important position and application value in social life. (1) One of the mathematical models, with flexible structure, various forms and efficient computing performance, the image processing variational model established in the early 1990s has become an important mathematical tool in the field of digital image processing, and has attracted extensive research and attention from scholars at home and abroad. Some key problems, such as denoising, restoration and segmentation, are discussed. In view of the poor adaptive performance of variational models in image processing and the easy occurrence of "ladder effect" in smooth regions, the characteristics and significance of variational models are studied, and the improvement ideas and methods are proposed. The main work of this paper is as follows: (1) Variational denoising in image processing. In the aspect of model research, an image denoising algorithm based on smoothed ROF model is proposed to solve the non-differentiable problem of regular terms in Rudin-Osher-Fatemi (ROF) model. The algorithm modifies the regular terms in the ROF model, which not only can get rid of the Courant-Friedrichs-Lewy condition, but also has the global convergence property. Compared with the time marching method and the fixed point method, each iteration updating corresponds to an original variable and a dual variable. The simulation results show that the proposed model can effectively reduce the image stationary region by choosing the appropriate smooth parameters. (2) In the research of variational model of image restoration, firstly, based on the analysis of total variational model, an improved total variational model of image restoration is proposed by using a more reasonable potential function. If there is noise in the image to be repaired, the noise can be removed from the outside of the information defect area at the same time. This not only avoids the "ladder effect" but also protects the important features of the image. Secondly, the solving process of Mumford-Shah model is complicated. An image restoration algorithm based on convex Mumford-Shah model is proposed. The algorithm takes full account of the different diffusion abilities of quadratic norm and total variational norm in the texture and smooth regions of the image, avoids the trouble of solving the length term of the curve, and uses split Bregman algorithm to solve the model quickly. (3) In the research of variational model of image segmentation, firstly, to solve the problem of high computational complexity of Mumford-Shah model, an image segmentation method based on constrained Mumford-Shah model is proposed. Degree term is used to transform the non-convex Mumford-Shah model into a convex optimization problem, and the pixel value of the image is always constrained within a reasonable range during the iteration process. Then the model is solved by alternating direction multiplier method. After obtaining the smooth solution of the model, the image is segmented by K-means clustering method, and the smooth and segmented image is realized. Secondly, when the coefficient matrix in the convex Mumford-Shah model is a random matrix, the efficiency of the alternating direction multiplier decreases, and an image segmentation algorithm based on the inexact alternating direction multiplier is proposed. Another advantage of this strategy is that if the number of segmentation targets is changed, the smooth solution of the image is not needed to be recalculated, so it is more convenient to apply. Simulation results show that the strategy can deal with the problem of image segmentation of degraded images, and the algorithm can also deal with the problem of image segmentation of degraded images. It has high segmentation accuracy.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TP391.41

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